TABLE OF CONTENTS In Chapter 3, we introduced quantum mechanics as the proper alternative to classical mechanics to describe physical phenomena, especially when the dimensions of the systems considered approach the atomic scale. The concepts we learned will now be applied to describe the physical properties of electrons in a crystal. During this process, we will make use of the simple quantum mechanical systems which were mathematically treated in the previous Chapter. This will lead us to the description of a very important concept in solid state physics namely that of the "energy band structures".
So far, we have discussed the energy spectrum of an electron in an atom, and more generally in a one-dimensional potential well. Modeling an electron in a solid is much more complicated because it experiences the combined electrostatic potential of all lattice ions and all other electrons. Nevertheless, the total potential acting on the electrons in a solid shares the symmetry of the lattice, and thus reflects the periodicity of the lattice in the case of a crystal. This simplifies the mathematical treatment of the problem and allows us to understand how the energy spectrum, wavefunctions and other dynamic characteristics (e.g. mass) of electrons in a solid are modified from the free particle case.
The Bloch theorem provides a powerful mathematical simplification for the wavefunctions of particles evolving in a periodic potential. The solutions of the Schr dinger equation in such a potential are not pure plane waves as...
